| L(s) = 1 | + 0.841·5-s + (1.16 + 2.37i)7-s − 3.64i·11-s + 5.53i·13-s + 0.841·17-s + 3.06i·19-s + 3.64i·23-s − 4.29·25-s + 8.89i·29-s − 7.82i·31-s + (0.979 + 1.99i)35-s − 3.29·37-s + 8.66·41-s − 2.32·43-s + 9.10·47-s + ⋯ |
| L(s) = 1 | + 0.376·5-s + (0.439 + 0.898i)7-s − 1.09i·11-s + 1.53i·13-s + 0.204·17-s + 0.704i·19-s + 0.760i·23-s − 0.858·25-s + 1.65i·29-s − 1.40i·31-s + (0.165 + 0.338i)35-s − 0.541·37-s + 1.35·41-s − 0.354·43-s + 1.32·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.734628137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734628137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.16 - 2.37i)T \) |
| good | 5 | \( 1 - 0.841T + 5T^{2} \) |
| 11 | \( 1 + 3.64iT - 11T^{2} \) |
| 13 | \( 1 - 5.53iT - 13T^{2} \) |
| 17 | \( 1 - 0.841T + 17T^{2} \) |
| 19 | \( 1 - 3.06iT - 19T^{2} \) |
| 23 | \( 1 - 3.64iT - 23T^{2} \) |
| 29 | \( 1 - 8.89iT - 29T^{2} \) |
| 31 | \( 1 + 7.82iT - 31T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 - 8.66T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 9.10T + 47T^{2} \) |
| 53 | \( 1 + 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 9.10iT - 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 0.354iT - 71T^{2} \) |
| 73 | \( 1 - 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 9.10T + 83T^{2} \) |
| 89 | \( 1 - 6.97T + 89T^{2} \) |
| 97 | \( 1 - 3.57iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.195445195379473649810143427219, −8.702760028851727254728756497590, −7.85116547520333542471725518690, −6.95676058914386624038922993339, −5.85119687776461848011802653216, −5.65899046497278388907924135802, −4.42958957248029778604759815975, −3.51002600121575796343761909010, −2.34117522286884327240587228942, −1.44796877008991506148539990967,
0.63058326394907823018197969918, 1.94083863196988844013294070885, 3.00481911452535751159468685466, 4.17029027787894079096542347561, 4.86414356674289326025725098732, 5.75376333816305434664700618884, 6.67687285489329053936231190385, 7.57988852286670573265156628865, 7.951468828362041176698699670773, 9.036248936314534441501316845718
